One-dimensional Graph Perturbations H.L. Wietsma

One-dimensional Graph Perturbations

H.L. Wietsma

A Bachelor Thesis

Submitted to the Faculty of Mathematics and Natural Sciences University of Groningen

Groningen, The Netherlands August 2006

Supervisors:

Prof. H.S.V. De Snoo

Prof. H. Winkler

iii

Preface and acknowledgements

First and foremost I would like to express my gratitude to my supervisor Henk de Snoo for providing the interesting subject of this thesis and for giving me the opportunity to get involved in doing research. In particular, by giving small sug- estions and lots of articles, perhaps a bit frustrating a first, my understanding of mathematics has grown immensly.

Secondly, I would like to thank Henrik Winkler for aiding me during my research.

I particular, when I got stuck he always knew how to clarify my problems.

Finally, I would like to thank Ineke Kruizinga for taking care of the administrative tasks related to this thesis.

Oldeboorn, August 2006, Hendrik Luit Wietsma

v

Contents

Preface and acknowledgements iii

Contents v

1 Introduction 1

2 Bordered matrices 2

3 Rational Nevanlinna functions 4

4 One-dimensional graph perturbation 5

5 Q_∞(z) 8

6 Eigenvalues of A(τ ) and representations of A 10

7 limiting behavior of Q_τ(z) 12

8 Graph perturbations and bordered matrices 15

A Existence and uniqueness proof 17

Bibliography 21

1

Chapter 1: Introduction

This report is concerned with one-dimensional graph perturbations denoted A(τ ), τ ∈ R ∩ {∞}, of a self-adjoint matrix A, where A(0) = A. These perturbations are studied by means of the Weyl functions associated with the self-adjoint matrices A(τ ), denoted by Q_τ(λ). In particular we will show that all these Weyl functions are rational Nevanlinna functions without a linear term, except for the special Weyl function Q_∞(λ). This Weyl function corresponds to a special one-dimensional per- turbation of A the so-called Friedrichs perturbation, which is not a matrix. We will show in detail how this linear term is created in the Weyl function Q∞(λ) by an escaping eigenvalue.

Furthermore, we will see that all the A(τ ) can be intrepreted as self-adjoint exten- sions of a so-called simple matrix S. Finally, using the concept of a bordered matrix we will show how we can describe the self-adjoint matrices A(τ ), τ ∈ R, and give a meaning to the special object A(∞).

The contents of this report are as follows; in Chapter 2 we introduce the concept of a bordered matrix In Chapter 3 we will recall some basic facts about rational Nevanlinna functions. In Chapter 4 we will introduce the one-dimensional graph perturbation and determine the associated Weyl functions. Chapter 5 is wholly devoted to the Weyl function corresponding to the special perturbation. In Chapter 6 we will look at the graph perturbations as extension of a simple matrix and in Chapter 7 we will show how the linear term is created in the special Weyl function Q_∞(λ). In the last chapter, Chapter 8, we will combine the theory to describe the self-adjoint matrices using bordered matrices. Finally, to make the report more or less self-contained, an appendix has been added which contains a number of results mainly related to rational Nevanlinna funtions.

2

Chapter 2: Bordered matrices

Two sequences of real numbers (λ_i)ⁿ₁ and (µ_i)ⁿ⁻¹₁ are called alternating if the fol- lowing inequalities are satisfied;

(2.1) λ₁ < µ₁< λ₂ < µ₂< . . . < µ_n−1< λ_n. Let A be the (n − 1) × (n − 1) matrix defined by;

A = diag {µ₁, . . . , µ_n−1}.

The question is if there exists a bordered matrix A of the following form;

(2.2) A = A g

g^∗ a

 ,

with σ(A) = {λ1, · · · λn}, a ∈ C and g ∈ Cⁿ⁻¹. If such a bordered matrix exists, then

(2.3) a =

n

X

i=1

λ_i−

n−1

X

i=1

µ_i,

due to a trace argument. Furthermore, Schur’s identity (see Appendix) gives the following representation for A − z;

(2.4) A − z =

 I 0

g^∗(A − z)⁻¹ I

 A − z 0 0 S(z)

 I (A − z)⁻¹g

0 I

 , where the Schur complement S(z) is given by

(2.5) S(z) = (a−z)−g^∗(A−z)⁻¹g = (a−z)−

n−1

X

i=1

g²_i

µ_i− z, z ∈ C\{µ₁, . . . , µ_n−1}.

The identity (2.4) shows that

(2.6) det(A − z) = S(z) det(A − z), z ∈ C \ {µ₁, . . . , µ_n−1} or, equivalently,

(2.7) S(z) = Πⁿ_i=1(λi− z)

Πⁿ⁻¹_i=1(µ_i− z), z ∈ C \ {µ1, . . . , µn−1}.

Chapter 2. Bordered matrices 3

Hence, equation (2.5) and (2.7) imply that g²_k= − Πⁿ_i=1(λi− µ_k)

Πⁿ⁻¹_i=1,i6=k(µi− µ_k), 1 ≤ k ≤ n − 1,

since g_k² is the residue of S(z) at µ_k. Note that the g²_k, as given by (2) are positive, since the sequences (λ_i)ⁿ₁ and (µ_i)ⁿ⁻¹₁ are alternating.

Proposition 2.1. Let a ∈ R and g_k∈ C, 1 ≤ k ≤ n − 1, be given by (2.3) and (2).

Define A by (2.2), then the spectrum of A is given by λ₁, . . . , λ_n.

Proof. The foregoing discussion shows that the Schur complement of A is given by (2.7). Equation (2.6) then shows that the eigenvalues of A are the λ_i’s, which completes the proof.

Let A and A be related as in (2.2), then A is recovered form A by a compression;

A = P_ω⊥A|_ω⊥, where ω = 0 . . . 0 1
T

∈ Cⁿ. To prove the above equality let f = f^′ f_n
T

be any element of Cⁿ, then P_ω⊥A|_ω⊥f =I_n−1 0

0 0

  A g g^∗ a

 f^′ 0



=Af^′ 0



=A 0 0 0



f = Af.

4

Chapter 3: Rational Nevanlinna functions

A function f (z) is called Nevanlinna if it has the following properties; f (z) is holo- morphic on C\R, f (z) = f (z) and Im (f (z))/Im (z) ≥ 0 for all z ∈ C\R. From these properties one can deduce that the sum and composition of Nevanlinna functions are Nevanlinna functions. A special class of Nevanlinna functions are the rational Nevanlinna functions, which have the following general form;

f (z) = m1

λ₁− z + m2

λ₂− z + . . . + mn

λ_n− z + αz + β

with α, β, λi, mi ∈ R, α ≥ 0 and mi> 0 1 ≤ i ≤ n.

(3.1)

A proof of this statement is given in the appendix. Rational Nevanlinna functions as in (3.1) with α = 0 have so-called moments; the zeroth respectively the first moment, denoted by M0 respectively M1, are defined by M0 = P_n

i=1mi and M₁ =P_n

i=1λ_im_i. Furthermore the following relation holds;

(3.2)

n

X

i=1

λi−

n−1

X

i=1

µi = M1

M₀,

where the µ_i are the zero’s of f (z). See also Lemma A.4 in the appendix.

Next look at the Schur complement of (A − z)⁻¹, it follows from (2.4) that

< (A − z)⁻¹ω, ω >= 1

S(z) = Πⁿ⁻¹_i=1(µi− z) Πⁿ_i=1(λ_i− z) =

n

X

i=1

m_i λ_i− z, where ω = 0 . . . 0 1
T

∈ Cⁿ and

m_k= Πⁿ⁻¹_i=1(µ_i− λ_k)

Πⁿ_i=1,i6=k(λ_i− λ_k), 1 ≤ k ≤ n.

Because the sequences (λ_i)ⁿ₁ and (µ_i)ⁿ⁻¹₁ are alternating the m_kare positive, therefor 1/S(z) is a rational Nevanlinna function. This is in agreement with the fact that

(3.3) −S(z) = z − a +

n−1

X

i=1

g²_i µ_i− z

is a rational Nevanlinna function, because the class of Nevanlinna functions is closed under composition and −1/z is Nevanlinna.

5

Chapter 4: One-dimensional graph perturbation

Let A be a n × n selfadjoint matrix with eigenvalues λ₁ < λ₂ < . . . < λ_n and let ω be a normalized element of Cⁿ. Then the one-dimensional graph perturbations of A, denoted by A(τ ), are given by;

(4.1) A(τ ) = A + τ < ·, ω > ω,

where τ ∈ R. Furthermore define the following two functions;

(4.2) γ(z) = (A − z)⁻¹ω and Q(z) =< (A − z)⁻¹ω, ω >, z ∈ C \ R.

γ(z) and Q(z) are called the γ-field respectively Weyl function of A. The following theorem shows the relation between the resolvents of A(τ ) and A;

Theorem 4.1.

(A(τ ) − z)⁻¹= (A − z)⁻¹− γ(z)< ·, γ(¯z) >

Q(z) + 1/τ, z ∈ C \ (σ(A) ∪ σ(A(τ ))).

Proof. Apply (4.1) minus z to the vector (A − z)⁻¹ω;

(4.3) (A(τ ) − z)(A − z)⁻¹ω = ω + τ < (A − z)⁻¹ω, ω > ω = (1 + τ Q(z))ω.

Dividing both sides by 1 + τ Q(z) and premultipling with (A(τ ) − z)⁻¹ we get;

(4.4) (A(τ ) − z)⁻¹ω = (A − z)⁻¹ω 1 + τ Q(z) . Next apply (4.1) minus z to (A − z)⁻¹k for arbitrary k ∈ Cⁿ;

(A(τ ) − z)(A − z)⁻¹k = (A − z)(A − z)⁻¹k + τ < (A − z)⁻¹k, ω > ω

= k + τ < k, γ(¯z) > ω

If the above identity is premultiplied by (A(τ ) − z)⁻¹ and (4.4) is used we have that (A − z)⁻¹k = (A(τ ) − z)⁻¹k + (A(τ ) − z)⁻¹ωτ < k, γ(¯z) >

= (A(τ ) − z)⁻¹k +(A − z)⁻¹ωτ < k, γ(¯z) >

1 + τ Q(z) from which the theorem follows.

6 Chapter 4. One-dimensional graph perturbation

To obtain a partial fraction expansion for the Weyl function of A use the fact that A is selfadjoint, which means that there exists a unitairy matrix U and a diagonal matrix D, with σ(D) = σ(A), such that A = U DU^∗. Using this representation of A we can determine Q(z);

Q(z) =< (A − z)⁻¹ω, ω >=< (U DU^∗− z)⁻¹ω, ω >

=< (U (D − z)U^∗)⁻¹ω, ω >=< U (D − z)⁻¹U^∗ω, ω >

=< (D − z)⁻¹U^∗ω, U^∗ω >=

n

X

i=1

m_i

λ_i− z, z ∈ C \ σ(A), (4.5)

where m_i =< (U^∗ω)_i, (U^∗ω)_i > is nonnegative. Note that this implies that (4.6)

n

X

i=1

m_i =< U^∗ω, U^∗ω >= ||U^∗ω||² = ||ω||² = 1.

We conclude that Q(z) in (4.5) is a rational Nevanlinna function. Similar to A, we define a Weyl function, Q_τ(z), for A(τ ) by

(4.7) Qτ(z) = −τ + < (A(τ ) − z)⁻¹p

τ²+ 1 ω,p

τ²+ 1 ω > .

Note that for τ = 0 we obtain Q(z) as in (4.2). The following relation exists between the Weyl functions Q(z) and Qτ(z).

Theorem 4.2.

(4.8) Q_τ(z) = Q(z) − τ

1 + τ Q(z), τ ∈ R ∪ {∞}

Proof. To prove this statement use the relation between (A(τ ) − z)⁻¹and (A − z)⁻¹ given by Theorem 4.1 in equation (4.7);

Q_τ(z) + τ =<



(A − z)⁻¹− γ(z) 1

Q(z) + 1/τ < ·, γ(¯z) >

p

τ²+ 1 ω,p

τ²+ 1 ω >

= (τ²+ 1)



< (A − z)⁻¹ω, ω > − < < ω, γ(¯z) >

Q(z) + 1/τ γ(z), ω >



= (τ²+ 1)Q(z)



1 − Q(z) Q(z) + 1/τ



= (τ²+ 1)Q(z) 1 + τ Q(z) It follows that

Q_τ(z) = −τ +(τ²+ 1)Q(z)

1 + τ Q(z) = Q(z) − τ 1 + τ Q(z).

The above result shows that Q_τ(z) is a rational Nevanlinna function if Q(z) is, because (4.8) can be written as;

(4.9) Q_τ(z) = 1

τ +τ²+ 1 τ²

−1 Q(z) + 1/τ,

Chapter 4. One-dimensional graph perturbation 7

and the class of Nevanlinna functions is closed under composition, addition and multiplication by a positive real number. Thus we can write Qτ(z) , τ 6= ∞, as;

(4.10) Q_τ(z) = α(τ )z + β(τ ) +

n

X

i=1

m_i(τ ) λ_i(τ ) − z,

where the λ_i(τ ) are the zero’s of Q(z) + 1/τ . Here α(τ ) and β(τ ) can both be determined by means of a limit;

α(τ ) = lim

z→∞

Q_τ(z)

z and β(τ ) = lim

z→∞(Q_τ(z) − a(τ )z) .

Since the limit of Q(z) as z tends to infinity is zero, see (4.5), Theorem 4.2 shows that α(τ ) = 0. Similarly, Theorem 4.2 shows that β(τ ) = −τ . Thus Q_τ(z) equals

Q_τ(z) = −τ +

n

X

i=1

m_i(τ ) λi(τ ) − z.

The m_i(τ ) can be determined by means of a residue argument;

mi(τ ) = −Res_λi(τ ) (Qτ(z) + τ ) = − lim

z↓λⁱ(τ )(z − λi(τ ))(Qτ(z) + τ ).

The last equality holds because Q_τ(z) has only poles of order 1. Use equation (4.2) in the above equation, combined with the fact that λi(τ ) is a zero of Q(z) + 1/τ .

m_i(τ ) = − lim

z↓λi(τ )(z − λ_i(τ )) Q(z) − τ 1 + τ Q(z)+ τ



= − lim

z↓λi(τ )

τ²+ 1 τ

Q(z)(z − λ_i(τ )) Q(z) + 1/τ

= lim

z↓λi(τ )

Q(z)(τ²+ 1) τ

z − λ_i(τ )

Q(z) − Q(λ_i(τ )) = τ²+ 1 τ²

1 Q^′(λ_i(τ )) (4.11)

Note that the above equation shows that the m_i(τ ) are positive, because Q^′(z) is positive on the real line where it is holomorphic (see Appendix Lemma A.2), this is in agreement with the fact Qτ(z) is Nevanlinna.

8

Chapter 5: Q

(z)

Define Q_∞(z) to be the limit of Q_τ(z) as τ tends to infinity. Use Theorem 4.2 to obtain the following expression for Q∞(z);

(5.1) Q_∞(z) = lim

τ →∞Q_τ(z) = lim

τ →∞

Q(z) − τ

1 + τ Q(z) = −1 Q(z).

The above relation reveals an important link, namely the zeros of Q_∞(z) are the poles of Q(z) and visa versa. Since Q_∞(z) is a composition of Nevanlinna functions, it is itself a rational Nevanlinna function and has a partial fraction expansion;

(5.2) Q_∞(z) = α(∞)z + β(∞) +

n−1

X

i=1

m_i(∞) µ_i− z,

where the µ_iare the zero’s of Q(z). Here α(∞) and β(∞) can be obtained by means of the following limits;

α(∞) = lim

z→∞

Q_∞(z)

z and β(∞) = lim

z→∞(Q_∞(z) − a(∞)z) . To determine α(∞) use expression (5.1) for Q_∞(z);

α(∞) = lim

z→∞

Q_∞(z) z = lim

z→∞

−1

Q(z)z = lim

z→∞

−1 zP_n

i=1 mⁱ λi−z

= lim

z→∞

1 Pn

i=1 mi

1−λⁱ/z

= 1

Pn

i=1m_i = 1 M₀ (5.3)

To determine β(∞) it is convenient to use an alternative expression for Q(z), there- fore use that the µi’s are defined to be the zeros of Q(z);

(5.4) Q(z) =

n

X

i=1

mi

λ_i− z = (Pn

i=1m_i)Qn−1

i=1(µ_i− z) Q_n

i=1(λ_i− z) . Use (5.4) to determine β(∞);

β(∞) = lim

z→∞(Q_∞(z) − a(∞)z) = lim

z→∞

−1 Q(z) − z

M0

= lim

z→∞

−Q_n

i=1(λ_i− z) M0Qn−1

i=1(µi− z) − zQ_n−1

i=1(µ_i− z) M0Qn−1

i=1(µi− z)

= lim

z→∞

(−1)ⁿ⁻¹ Pn−1

i=1 µ_i−Pn i=1λ_i

zⁿ⁻¹+ o(zⁿ⁻²) M₀(−1)ⁿ⁻¹zⁿ⁻¹+ o(zⁿ⁻²)

= P_n−1

i=1 µ_i−P_n

i=1λ_i

M₀ = −M₁

M²₀ , (5.5)

Chapter 5. Q_∞(z) 9

where the last equality holds because of (3.2). Equations (5.3) and (5.5) show that Q∞(z) is equal to

Q_∞(z) = z

M₀ −M₁ M²₀ +

n−1

X

i=1

m_i(∞) µ_i− z.

To obtain an expression for the m_i(∞) use, once more, a residue argument;

(5.6) m_i(∞) = lim

z↓µi

Q_∞(z)(z − µ_i) = lim

z↓µi

z − µ_i

Q(z) − Q(µi) = 1 Q^′(µi).

Note that the m_i(∞) are positive, furthermore the equations for the m_i(∞) seem consistent with the earlier result for m_i(τ ), see (4.11).

10

Chapter 6: Eigenvalues of A (τ ) and representations of A

Let S be defined as

(6.1) S = A|_ω⊥,

where A is a selfadjoint operator and ω ∈ H \ {0}. Here H is the Hilbert space on which S and A are defined. In that case A is called a selfadjoint extension of S, note that all A(τ ) as defined in (4.1) are extensions of the same S.

If S has an eigenvalue λ with eigenvector e, then e ∈ ω^⊥ and Ae = Se = λe,

thus e is also an eigenvector for A. Thus in the case that S contains an eigenspace the self-adjoint extensions of S contain the same space. Therefor divide the domain of S, H, into two parts, Hr, the space which consists of the eigenvectors of S and into a remainder H_s.

Thus by restricting the domain we obtain an operator which has no eigenvectors, such an operator is called simple or completely non-selfadjoint. Since the above decomposition of H is always possible, without loss of generality S will be assumed to be simple from now on. The property of being simple can also be characterized with respect to A and ω.

Proposition 6.1. The operator S = A|_ω⊥ is simple if and only if no eigenvector of A is orthogonal to ω. In this case, the spectrum of A is necessarily simple, i.e. each eigenvalue of A has multiplicity 1.

Proof. The existence of an invariant subspace of A in H ⊖ span {ω} is equivalent to the existence of eigenvectors of A in H ⊖ span {ω}, which shows the first statement.

Assume that λ is an eigenvalue with multiplicity 2, and let e₁ and e₂ be linearly independent eigenvectors. If (e1, ω) = c1 6= 0 and (e2, ω) = c26= 0, then

e₁/c₁− e₂/c₂ 6= 0

is an eigenvector belonging to the eigenvalue λ₀, s.t. (ω, e₁/c₁− e₂/c₂) = 0.

With the above characterization of eigenvalues of selfadjoint extensions it is possible to prove that A and A(τ ), τ 6= 0, do not have an eigenvalue in common. In that case Theorem 4.1 indicates that the eigenvalues of A(τ ) are the zero’s of Q(z)+ 1/τ ,

Chapter 6. Eigenvalues of A(τ ) and representations of A 11

since the resolvent of A(τ ) is not holomorphic at those points. Furthermore Theorem 4.2 then shows that the eigenvalues of A(τ ) are the zeros of Qτ(z), similar to the eigenvalues of A.

Proposition 6.2. If A and A(τ ), τ 6= 0, are selfadjoint extensions of a simple operator S as in (6.1) which are related as in (4.1), then A(τ ) and A do not have an eigenvalue in common.

Proof. Suppose A and A(τ ) do have an eigenvalue, λ, in common, then there exist e, e(τ ) ∈ H, not necessarily distinct, such that

Ae = λe and A(τ )e(τ ) = λe(τ ).

In that case (4.1) indicates that

< (A(τ ) − λ)e(τ ), e > =< (A − λ)e(τ ), e > +τ < e(τ ), ω >< ω, e >

=< e(τ ), (A − λ)e > +τ < e(τ ), ω >< ω, e >, which is equivalent to

0 = τ < e(τ ), ω >< ω, e > .

If < e(τ ), ω >= 0. then it follows from (4.1), that e(τ ) is an eigenvector of A for λ and thus equal to e. Thus the above equation is in contradiction with Proposition 6.1.

Corollary 6.3. If A(τ ) is defined as in (4.1) and Q(z) as in (4.2), then the eigen- values of A(τ ) are the zero’s of Q(z) + 1/τ .

Since the function Q(z) determines the eigenvalues of A(τ ) for all values of τ , it is interesting to see if the representation of A has influence on Q(z) and thus on the eigenvalues of A(τ ).

Let B be an operator which is unitary equivalent to A, that means there exists an unitary transformation Q s.t. B = Q^∗AQ. Then

Q_B(z) =< (B − z)⁻¹ω^′, ω^′ >=< (Q^∗AQ − z)⁻¹ω^′, ω^′ >=< ((A − z)⁻¹Qω^′, Qω^′ >

The above function is equal to Q(z) if and only if ω^′ = Q^∗ω. In that case ω^′ is also normalized, since ||Q^∗ω|| = ||ω||, ω^′. Thus if a pair (A, ω) defines A(τ ) as in (4.1), the pair (B, ω^′) defines a B(τ ) which is unitary equivalent to A(τ ) for all τ ∈ R if and only if ω^′ = Q^∗ω. Thus different representations of A give the same perturbation if and only if ω is adjusted accordingly.

12

Chapter 7: limiting behavior of Q

(z)

To show that Q_τ(z) is consistent with Q(z) and Q_∞(z), it is necessary to know how the eigenvalues vary as functions of τ .

Theorem 7.1. Let λ1, . . . , λnbe the poles of Q(z), in increasing order. Likewise let µ₁, . . . , µ_n−1be the zeros of Q(z) in increasing order and let λ₁(τ ), . . . , λ_n(τ ) be the zero’s of Q(z) + 1/τ in increasing order. Then the λ_i(τ )’s are increasing functions of τ ,

τ →0limλi(τ ) = λi 1 ≤ i ≤ n,

τ →∞lim λ_i(τ ) = µ_i 1 ≤ i ≤ n − 1 and λ_n(τ ) tends to infinity as τ tends to infinity

Proof. Note first that λ_k ≤ µ_k ≤ λ_k+1 for any k, because since Q(z) is continuous and Q^′(z) is positive (see Lemma A.2) on R \ {λ₁, . . . , λ_n} there has to be a zero between any two poles of Q(z).

Since Q(λ_i(τ )) ≤ 0, if τ > 0, and Q(z) is increasing it follows that λ_i ≤ λ_i(τ ) ≤ µ_i. Furthermore

dQ(λ_i(τ ))

dτ = d(1/τ )

dτ ⇔ Q^′(λ_i(τ ))λ^′_i(τ ) = 1

τ² ⇔ λ^′_i(τ ) = 1 τ²Q^′(λ_i(τ )), from which we conclude that the λi(τ ) are increasing as functions of τ . Since there are n − 1 zero’s λ_n(τ ), being larger than all the zero’s, must be unbounded.

Corollary 7.2. The following inequalities hold for all τ ∈ R⁺∪ {∞}

λ₁(τ ) ≤ µ₁≤ λ₂(τ ) ≤ . . . ≤ µ_n−1 ≤ λ_n(τ ) Strict inequalities hold in the case that τ ∈ R⁺.

To prove that Q_τ(z) converges to Q_∞(z), we need to know how the linear term in Q_∞(z) is created. The fact that λ_n(τ ) tends to infinity as τ tends to infinity gives a clue has how that term is created. To give an exact proof we need the following lemma.

Lemma 7.3.

n

X

i=1

λ_i(τ ) =

n

X

i=1

λ_i+ τ

n

X

i=1

m_i

Chapter 7. limiting behavior of Q_τ(z) 13

Proof. Use the fact that the trace of a matrix is the sum of its eigenvalues in (4.1) combined with (4.6);

tr(A(τ )) = tr(A + τ < ·, ω > ω) = tr(A) + τ tr(< ·, ω > ω)),

which is equivalent to

n

X

i=1

λ_i(τ ) =

n

X

i=1

λ_i+ τ ||ω||² =

n

X

i=1

λ_i+ τ

n

X

i=1

m_i.

Theorem 7.4. If m_n(τ ) is defined as in (4.11), λ_n(τ ) as in Theorem 7.1 and M₀ and M₁ as before, then

τ →∞lim

m_n(τ )

λ_n(τ ) − z − τ = αz + β, where α = 1

M₀ and β = −M₁ M²₀ . Proof.

τ →∞lim

m_n(τ )

λ_n(τ ) − z − τ = lim

τ →∞

m_n(τ ) − τ (λ_n(τ ) − z)

λ_n(τ ) − z = lim

τ →∞

τ²+1 τ²

1

λⁿ(τ )Q^′(λⁿ(τ ))− τ + _λ^τ

n(τ )z 1 −_λn^z_{(τ )}

= lim

τ →∞

(τ²+1)λⁿ(τ )

τ²λn(τ )²Q^′(λn(τ ))− τ^τ_τ²2λ^λn^{n(τ )}(τ )^22QQ^′′(λ^(λnn^{(τ ))}(τ ))+_λn^τ_{(τ )}z 1 −_λ ^z

n(τ )

= lim

τ →∞

λⁿ(τ )−τ λⁿ(τ )²Q^′(λⁿ(τ ))

λⁿ(τ )²Q^′(λⁿ(τ )) +_τ2 ^{λn(τ )}

λⁿ(τ )²Q^′(λⁿ(τ ))+ _λ^τ

n(τ )z 1 −_λn^z_{(τ )}

(7.1)

Since the derivative of Q(z) equals;

(7.2) Q^′(z) = m₁

(λ₁− z)² + m₂

(λ₂− z)² + . . . + m_n (λ_n− z)²,

it is clear that by the asymptotic behavior of λ_n(τ ), see Theorem 7.1, that λ_n(τ )²Q^′(λ_n(τ )) tends to M0 =Pn

i=1mi. Therefore by Lemma 7.4 the second term in the numer- ator of (7.1) goes to zero. Furthermore the indicated lemma proves that the last term in the numerator goes to _M¹

0z.

Therefore only the first term of the numerator needs to be investigated. To be able

14 Chapter 7. limiting behavior of Q_τ(z)

to take the limit τ will be eliminated by using Lemma 7.4.

τ →∞lim

λ_n(τ ) − τ λ²(τ )Q^′(λ_n(τ )) λ_n(τ )²Q^′(λ_n(τ ))

= lim

τ →∞

λ_n(τ ) −_M¹

0

λ_n(τ ) +Pn−1

i=1 λ_i(τ ) −Pn i=1λ_i

λ²(τ )Q^′(λ_n(τ )) λn(τ )²Q^′(λn(τ ))

= lim

τ →∞

λ_n(τ ) Pn

i=1m_i−Pn

i=1 mi

λⁱ−λⁿ(τ ))

 M0λn(τ )²Q^′(λn(τ )) − lim

τ →∞

P_n−1

i=1 λ_i(τ ) −P_n

i=1λ_i M0

= lim

τ →∞

λ_n(τ )P_n

i=1m_i^(λi−λ_λ^{n(τ ))2−λn(τ )2}

i−λⁿ(τ ))

M₀λ_n(τ )²Q^′(λ_n(τ )) − P_n−1

i=1 µ_i−P_n

i=1λ_i M₀

= lim

τ →∞

−2Pn

i=1m_iλ_i(_λ^{λn(τ )2}

i−λn(τ ))+Pn

i=1m_iλ²_i(_λ ^{λn(τ )}

i−λn(τ ))

M₀λ_n(τ )²Q^′(λ_n(τ )) + M1

M²₀

= −2Pn i=1m_iλ_i M₀M₀ +M1

M²₀ = −2M1

M²₀ +M1

M²₀ = −M1

M²₀,

where the two expressions for M₁ as given in Chapter 3 are used.

To prove that the fraction expansion of Q_τ(z) converges to Q_∞(z), it is enough by the results in Theorem 7.1 and 7.4 to show that the m_k(τ ) as in (4.11) converge to the m_k(∞) as in (5.6), but that follows directly from Theorem 7.1.

Similarly, to prove that the fraction expansion of Q_τ(z) converges to Q(z), we only need to show that the m_i(τ ) converge to the m_i, for which we will use the following lemma.

Lemma 7.5.

limτ ↓0

τ

λ_k(τ ) − λ_k = 1 m_k Proof. λ_k(τ ) is a zero for Q(z) + 1/τ , thus

−1 = τ Q(λ_k(τ )) =

n

X

i=1

τ mi

λ_i− λ_k(τ ).

Because the λ_i are all different and lim_{τ →0}λ_k(τ ) = λ_k, the lemma is proven.

Determine the limit of the m_i(τ ) by the use of the above lemma;

(7.3) lim

τ ↓0m_i(τ ) = lim

τ ↓0(τ²+ 1) 1 Pn

i=1 τ²mi

(λi−λi(τ ))²

= 1

m_ilim_{τ ↓0}(_λ ^τ

i(τ )−λⁱ)² = m_i. This proves the convergence of Qτ(z) when τ goes to zero.

15

Chapter 8: Graph perturbations and bordered matrices

After looking at bordered matrices in the Chapter 2, the following chapters have been used to investigate graph perturbations, where the perturbations have been studied by means of their Weyl functions. In Chapter 6 we saw that this one- dimensional family could in fact be seen as the selfadjoint extensions of a simple operator S. In this chapter those parts are combined, so the one-dimensional family of selfadjoint extensions will be described by means of bordered matrices.

Every graph perturbation as in (4.1) of a simple operator S is characterized by a selfadjoint n × n matrix A, a normalized perturbation ω ∈ Cⁿ and a parameter τ ∈ R ∪ {∞}. The corresponding Weyl function, Q(z), is a rational Nevanlinna function. By Corollary 7.2 the sequence of poles of Q(z), (λ_i)ⁿ₁, and the sequence of zeros of Q(z), (µ_i)ⁿ⁻¹₁ , are alternating.

If we define A = diag {µ₁, · · · , µ_n−1}, then by Proposition 2.1 there exists a unitary transformation P of A such that

A = P AP˜ ^∗ = A g g^∗ a

 ,

where g and a are as in (2) respectively (2.3).

Furthermore, if we define ˜ω = 0 . . . 0 1
T

∈ Cⁿ, then it was deduced in the Chapter 3 that < ( ˜A − z)⁻¹ω, ˜˜ ω >= Q(z). The uniqueness part of Chapter 6 then allows us to conclude that ˜ω = P^∗ω and that ˜A + τ < ·, ˜ω > ˜ω is unitary equivalent to A(τ ) for all τ . Thus all one-dimensional perturbations of A are unitary equivalent to

(8.1) A + τ < ·, ˜˜ ω > ˜ω = A g g^∗ a + τ

 ,

Thus using bordering matrices we get a simple representation for all the one- dimensional extensions of S. We see that if τ tends to infinity in (8.1), the ma- trix does not represent an operator anymore. But if its domain is restricted to (span {ω})^⊥ we do have an operator, namely S. So the question is what happens to a vector ξ = 0 . . . 0 ξ_n
∈ span {ω}. For τ ∈ R (8.1) indicates that ξ/τ is sent to ^g_τ ^a_τ + ξ_n
, if we sent τ to infinity this means that zero is mapped to the entire span of ω. We conclude that A(∞) is not an operator, but represents a so-called multivalued operator or relation. Thus (8.1) can be given meaning for τ = ∞ by

16 Chapter 8. Graph perturbations and bordered matrices

defining;

A(∞) = A|_{(span {ω})}⊥+ span {ω}.ˆ

In that case A(∞) has eigenvalues µ1, . . . , µn−1 and the eigenvalue ∞ can be seen as creating the multivalued part.

We conclude that the one dimensional family of selfadjoint extensions of a simple operator S as in (6.1) has a simple representation if bordered matrices are used.

Furthermore their is one special extension which not an operator, but a relation.

17

Appendix A: Existence and uniqueness proof

We begin by a well-known proposition which can be verified immediately.

Proposition A.1. Let A, B, C and D be p × p, p × q, q × p and q × q matrices respectively. If A is invertible, then the following identity, the so-called Schur’s matrix identity, holds;

A B C D



=

 I_p 0

CA⁻¹ Iq

 A 0

0 D − CA⁻¹B

 Ip A⁻¹B 0 Iq

 . The expression D − CA⁻¹B is called the Schur complement.

Next we consider Nevanlinna functions. A function f belongs to the class of Nevanlinna functions, denoted by N, if f is holomorphic on C \ R, f (z) = f (z) and Im (f (z))/Im (z) ≥ 0, for all z ∈ C \ R.

A classic result is that the property of being a Nevanlinna function is equivalent to admitting an integral representation of the following form;

(A.1) f (z) = αz + β +

Z

R

 1

t − z − t t²+ 1

 dσ(t),

where α ≥ 0, β ∈ R and z ∈ C \ R. Furthermore σ, which is called the spectral measure of f , is a non-decreasing function which satisfies the following integrability criterium;

Z

R

dσ(t) t²+ 1 < ∞.

A property of the class of Nevanlinna functions is that it is closed under composition and addition, which follows directly from the definition. The behavior of Nevanlinna functions on the real line is simple, as is shown by the following lemma adapted from Donoghue, cf. [1].

Lemma A.2. On intervals (a, b) ∈ R where f is holomorphic, f is a non-decreasing function. If additionally f is non-constant, then f is a strictly increasing.

Proof. Since f (z) = f (z) it is clear that f (z) = φ(z) + iψ(z) is real on the real line.

In the upper half plane f (z) can only be real if f (z) is constant. For suppose it is real outside the real line, then ψ has a zero. Since ψ, and φ, are harmonic ψ has to be identically zero due to the mean value theorem for harmonic functions, see [2]. If ψ is constant, then φ is also constant due to the Cauchy-Riemann equations.

Thus a non-constant Nevanlinna function is real only on the real line.

18 Appendix A. Existence and uniqueness proof

Let (a, b) ∈ R be an interval on which f is holomorphic, then ψ(ξ) = 0, for ξ ∈ (a, b).

Let 0 < η ∈ R then ψ(ξ + iη) > 0, since f (z) is not real inside the upper half plane.

So ψ(ξ + iη) − ψ(ξ) > 0 if η > 0 and similar if η < 0 then ψ(ξ + iη) − ψ(ξ) < 0, thus ψ_y(ξ) > 0. The Cauchy-Riemann equations then indicate that φ_x(ξ) = ψ_y(ξ) > 0, which proves the statement.

There are several subclasses of Nevanlinna functions, denoted by N_r. If r ∈ (0, 2), the condition is that α as in (A.1) is zero and

Z

R

dσ(t)

|t|^r+ 1 < ∞.

If −r ∈ N, then the condition on the spectral function is that Z

R

|t|^−rdσ(t) < ∞.

Clearly N_s⊂ N_t if s < t. Note that if f ∈ N₁ then _t2+1^t is integrable. Define β^′ = β −

Z

R

dσ(t) t²+ 1, then equation (A.1) simplifies to

f (z) = β^′+ Z

R

dσ(t) t − z.

Write _t−z¹ as a geometrical series, see [3],which is possible if |t/z| < 1, so z needs to be large, to get the following result;

Z

R

1

t − zdσ(t) = Z

R

1

−z 1

1 − t/zdσ(t) = −1 z

Z

R

∞

X

i=1

(t

z)ⁱdσ(t) = − Z

R

∞

X

i=1

tⁱ

zⁱ⁺¹dσ(t).

If f ∈ N_r, where r ≤ 0, then one can split off part of the summation,

f (z) = β^′− Z

R

1

zdσ(t) − · · · − Z

R

t^r

z^r+1dσ(t) − Z

R

∞

X

i=r+1

tⁱ

zⁱ⁺¹dσ(t).

Define the so-called global moments by

(A.2) M_k=

Z

R

t^kdσ(t),

then f has the following asymptotic expansion around ∞, if f ∈ N_r;

f (z) = β^′−

r

X

i=0

M_i

zⁱ⁺¹+ o( 1 z^r+1).

Appendix A. Existence and uniqueness proof 19

The global moments can be calculated by means of the following recursive formula;

M_k= − lim

z→∞(

k−1

X

i=0

z^k−iMi+ z^k+1(f (z) − β^′)).

Finally, a special class of Nevanlinna functions will be treated, namely the rational Nevanlinna functions. The following theorem shows that such Nevanlinna functions have a simple structure.

Theorem A.3. Each complex rational Nevanlinna function is of the following form;

f (z) = c₁

λ₁− z + c₂

λ₂− z + . . . + c_n

λ_n− z + αz + β with α, β, λ_i, c_i∈ R and α, c_i≥ 0.

Proof. The partial fraction theorem states that each rational function can be ex- pressed in the following way,

f (z) = R(z)/Q(z) =

r

X

i=1 n

X

j=1

cij

(λ_i− z)^j + P (z)

with P (z) a polynomial. Define C⁺ = {z ∈ C : Im(z) ≥ 0} and let f (z) be a complex rational function, then because of the partial fraction expansion and the fact that f (z) = f (z), all coefficients are real.

The next step is to look at the degree of the polynomial part and the degree of the fractions. Suppose that the degree of P (z) is m > 1, then since the highest degree dominates f (z) the image of ρe^2πk, k ∈ (0,^2π_m) and ρ ∈ R, dominates the image of f for large ρ, but this image contains the whole circle, a contradiction.

Similarly, one can show that the degree of the factors _(α ¹

i−z)^m must be one, because this term dominates f (z) in a small semi-circle around αi.

The only thing left to show is that the c_ij’s and a are nonnegative. Assume a < 0, take the point ρi ∈ C⁺ where ρ ∈ R is chosen such that the image of this point under az dominates f (z). Then aρi /∈ C⁺ dominates the image, a contradiction.

We have the same contradiction when a c_ij is not positive, by taking a point α_i+_|ρ|ⁱ , with ρ sufficiently small.

One important thing which this theorem makes clear is that a rational Nevanlinna function minus its linear part always has an asymptotic expansion, because its spectral measure is a discrete measure.

Finally, we consider in more detail the zeroth and first moments of a Nevanlinna function, see (A.2). As a byproduct we obtain a relation between the zeros and poles of Q(z) in terms of these two moments.

20 Appendix A. Existence and uniqueness proof

Lemma A.4. The zeroth respectively first moment of Q(z) as in (4.5), are given by M0 =P_n

i=1mi respectively M1 =P_n

i=1λimi. Furthermore the following relation holds

(

n

X

i=1

λ_i−

n−1

X

i=1

µ_i) = M₁ M0

. Proof. Calculate the zeroth and first moment for Q(z);

M₀= − lim

z→∞zQ(z) = − lim

z→∞

n

X

i=1

m_iz λ_i− z =

n

X

i=1

m_i

M1= − lim

z→∞zM0+ z²Q(z) = − lim

z→∞

n

X

i=1

m_i(z(λ_i− z) + z²)

λ_i− z =

n

X

i=1

λimi

This already gives us the first part of the lemma, to prove the second part of the lemma calculate the moments using (5.4);

M0 = − lim

z→∞zQ(z) = − lim

z→∞(−1)ⁿ⁻¹(

n

X

i=1

mi)z Qn−1

i=1(z − µ_i) Qn

i=1(λ_i− z) = (

n

X

i=1

mi)

M₁ = − lim

z→∞(zM₀+ z²Q(z)) = − lim

z→∞(M₀z + z²(−1)ⁿ⁻¹M₀ Q_n−1

i=1(z − µ_i) Qn

i=1(λ_i− z))

= M0 lim

z→∞

z²(−1)ⁿQn−1

i=1(z − µ_i) − zQn

i=1(λ_i− z) Q_n

i=1(λ_i− z)

= M₀ lim

z→∞

(−1)ⁿzⁿ(Pn

i=1λ_i−Pn−1

i=1 µ_i) + o(zⁿ⁻¹) Q_n

i=1(λ_i− z) = M₀(

n

X

i=1

λ_i−

n−1

X

i=1

µ_i).

21

Bibliography

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[2] S. Lang, ”Complex Analysis”, Springer-Verlag, Berlin-Heidelberg-New York, 1999

[3] S. Hassi, H.S.V. de Snoo, A.D.I. Willemsma, ”Smooth rank one pertubations of selfadjoint operators”, Proc. Amer. Math. Soc., 126 (1998), 2663-2675 [4] H. Dym, ”J contractive matrix functions, reproducing kernel Hilbert spaces

and interpolation”, American Mathematical Society, 1989

[5] S. Hassi, A. Sandovici, H.S.V. de Snoo, H. Winkler, ”Spectral gaps and one- dimensional perturbations”, to appear